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Let α and β (α > β) be the roots of the equation x2 - 8x + q = 0. If α2 - β2 = 16, then what is the value of q?

Let α and β (α > β) be the roots of the equation x2 - 8x + q = 0. If α2 - β2 = 16, then what is the value of q? Correct Answer 15

Concept:

The Standard Form of a Quadratic Equation is ax2 + bx + c = 0

Sum of roots = −b/a

Product of roots = c/a

Formulae 

(α - β)2 + 4.α.β = (α + β)2 

Calculation:

Given:

x2 - 8x + q = 0 and α2 - β2 = 16

Sum of roots = α + β = −b/a = -(- 8) = 8      ------(i)

Product of roots  = α.β = q      ------(ii)

We have α2 - β2 = 16

⇒ (α + β)(α - β) = 16

⇒ 8 × (α - β) = 16

⇒ (α - β) = 2      ------(iii)

We know that, (α - β)2 + 4.α.β = (α + β)2 

Putting the value of equation (i), (ii) and (iii), we get,

⇒ 22 + 4q = 82

⇒ 4q = 64 - 4 = 60

⇒ q = 15

∴ The value of q is 15

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